ε-ENTROPY OF ELECTROMAGNETIC FIELD

Maria Antonia Maisto

Università della Campania L. Vanvitelli

Inverse electromagnetic problem •  Inverse source

•  Inverse scattering

),( ωχ TrScatterer

Source Observation

),( ωRrE Ø  Antennas diagnostics Ø Antenna synthesis

Observation

),( ωRrE Ø  Geophysical survey Ø  Radar Imaging Ø  Medical Imaging Ø  Subsurface imaging

Operatorial point of view

Scattering/Radiation operator

•  Ill-posed problem in sense of Hadamard

Approximate solutions resulting from

compromise between accuracy and stability

Regularization method

Ef

•  Compact operator

•  Diffraction Theory

spot Observation angular sector Ω Object plane

of area A

Image plane‏

•  Sampling Approach

NDF = number of samples within the observation range

•  Operator spectrum approach

Observation domain J E

Source

JSource E

Observation domain

NDF = number of significant singular values

Number of Degrees of Freedom

Resolution

Raylight ‘s criterion Poynt-spread function

•  Regularization method

What is the maximum number of distinguishable messages?

Metric Entropy

Number of singular values NDF

1g

2g

σ2β

σ1β

3gσ3β

3f1f

2f

20

2 log βσ

=

≥ ≥ ∑N

n

nC H

Ú

Ú Ú Ú

Summary

Ø  Metric entropy estimation in terms of configuration parameters How to probe the field in order to maximize the entropy ? Ø  Example 1: Inverse source in near reactive zone

Ø  Example 2: Inverse scattering: multi-view configuration

Ø  Example 3: Inverse scattering: multi-frequency configuration

Ø  Conclusions

Metric entropy estimation

Estimation of an lower bound for metric entropy in terms of the configuration parameters

Singular values estimation σ n

σ̂ σ σ≤≤ %n n n multi-step behaviour

ˆ≥H HÚ ÚIn terms of the configuration parameters

If are known in closed form σ n If are not known

in closed form σ n

Example 1: Inverse source

Unbounded

2 2: ( ) ( )∈ → ∈u J L SD E LA RMagnetic source

The evanescent waves introduce a shaping of the singular values

oz

SCALAR, TWO DIMANSIONAL GEOMETRY

Example 1: Metric entropy estimation

/ 20λ=oz

Blue line refers to ε-entropy and the red one to its lower bound estimation.

Analytical estimation for the metric entropy in terms of the noise (ε) and probing distance zo

Examples 2 and 3: Inverse Scattering

Far zone

Strip object

∈Ωi

∈ΩoTwo strategy of illumination : •  Multi-view illumination •  Multi-frequency illumination

Examples 2 and 3: Singular values

•  The case of six angles of incidence

•  The case of three frequency

Singular value behavior of the scattering operator for , , and

View diversity Frequency diversity

The singular value of the scattering operator for and

The singular values can be approximated in closed form

•  View Diversity •  Frequency Diversity

Examples 2 and 3: Metric Entropy

Analitical estimation for the metric entropy in terms of the scattering parameters

•  Suppose to fix , , and

Comparison between diversities (1)

Single view/single frequency configuration Multi-view Configuration Multi-frequency configuration

M : number of diversities maxk : maximum frequency of illumination

Entropy increases with M

Comparison between diversities (2)

Suppose to fix , , and

•  View Diversity Multi-view configuration Single view configuration

•  Frequency Diversity Multi-frequency configuration Single frequecy configuration

The resolution remains unchanged !

Comparison between diversities (3)

Blue and red line refer to the normalized Psf for multi-view configuration when and , respectively.

1maxk 90m−=

1maxk 50m−=

Metric entropy for the multiview configuration.

1maxk 50m−=

1maxk 90m−=

The Entropy decreases and Resolution increases

Conclusions Kolmogorov entropy has been estimated by

analytical arguments

u2

u1

u3

v1

v2

v3

Kolmogorov entropy and Resolution can behave differently

[1] R Solimene, M A Maisto, R Pierri “Inverse scattering in presence of a reecting plane ” J. Opt. 18 (2) (2015) [2] R Solimene, M A Maisto, R Pierri “Inverse source in the presence of a reecting plane for the strip case”, J. Opt. Soc. Am. A 31 (12), 2814-2820 (2014) [3] R Solimene, M A Maisto, R Pierri “Role of diversity on the singular values of linear scattering operators: the case of strip objects” J. Opt. Soc. Am. A 30 (11), 2266-2272 (2013) [4] R Solimene, M A Maisto, G Romeo, R Pierri “On the singular spectrum of the radiation operator for multiple and extended observation domains” International Journal of Antennas and Propagation 2013 [5] Maria Antonia Maisto, Raffaele Solimene, Rocco Pierri “Metric entropy in linear inverse scattering” Advanced Electromagnetics 5, ( 2), 2016

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